a line segment has at most one midpoint

Theorem 1.

Proof.

Let [p,q] be a closed line segment and suppose m and m′ are midpoints.
If m:p:q then [m,p]<[m,q] so m is not a midpoint. Similarly we cannot have
p:q:m, so we have p:m:q. And also, p:m′:q. Suppose m≠m′. Without loss of
generality we can assume p:m:m′ and m:m′:q. But then [p,m′]>[p,m]≅[m,q]>[m′,q] so that
[p,m′]≇[m′,q], a contradiction. Hence m=m′.
∎